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Find the value of [tex]\cos 2\theta[/tex].

Given that [tex]\theta[/tex] is in the first quadrant and [tex]\sin \theta = \frac{12}{13}[/tex].

Double-Angle Formulas:
[tex]\[
\sin(2A) = 2 \sin A \cos A
\][/tex]
[tex]\[
\cos(2A) = \cos^2 A - \sin^2 A = 1 - 2 \sin^2 A = 2 \cos^2 A - 1
\][/tex]
[tex]\[
\tan(2A) = \frac{2 \tan A}{1 - \tan^2 A}
\][/tex]

Calculate [tex]\cos 2\theta[/tex]:
[tex]\[
\cos 2\theta = -\underline{[?]}
\][/tex]

Sagot :

To find the value of [tex]\(\cos 2\theta\)[/tex] given that [tex]\(\sin \theta = \frac{12}{13}\)[/tex] and that [tex]\(\theta\)[/tex] is in the first quadrant, we will use the double-angle formula for cosine and some trigonometric identities. Let's go through the solution step by step:

1. Given Information:
- [tex]\(\sin \theta = \frac{12}{13}\)[/tex]
- [tex]\(\theta\)[/tex] is in the first quadrant, implying that both [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex] are positive.

2. Find [tex]\(\cos \theta\)[/tex]:
- We use the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
Given [tex]\(\sin \theta = \frac{12}{13}\)[/tex], we first find [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[ \sin^2 \theta = \left(\frac{12}{13}\right)^2 = \frac{144}{169} \][/tex]
- Next, solve for [tex]\(\cos^2 \theta\)[/tex]:
[tex]\[ \cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{144}{169} = \frac{169}{169} - \frac{144}{169} = \frac{25}{169} \][/tex]
- Since [tex]\(\theta\)[/tex] is in the first quadrant, [tex]\(\cos \theta\)[/tex] is positive:
[tex]\[ \cos \theta = \sqrt{\cos^2 \theta} = \sqrt{\frac{25}{169}} = \frac{5}{13} \][/tex]

3. Apply the Double-Angle Formula for Cosine:
- The formula for [tex]\(\cos 2\theta\)[/tex] is:
[tex]\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \][/tex]
- Plug in the values we found for [tex]\(\cos \theta\)[/tex] and [tex]\(\sin \theta\)[/tex]:
[tex]\[ \cos^2 \theta = \left(\frac{5}{13}\right)^2 = \frac{25}{169} \][/tex]
[tex]\[ \sin^2 \theta = \left(\frac{12}{13}\right)^2 = \frac{144}{169} \][/tex]
- Substitute these into the double-angle formula:
[tex]\[ \cos 2\theta = \frac{25}{169} - \frac{144}{169} = \frac{25 - 144}{169} = \frac{-119}{169} \][/tex]

4. Conclusion:
- Therefore, the value of [tex]\(\cos 2\theta\)[/tex] is:
[tex]\[ \cos 2\theta = -\frac{119}{169} \approx -0.7041 \][/tex]

Thus, the value of [tex]\(\cos 2\theta\)[/tex] given [tex]\(\sin \theta = \frac{12}{13}\)[/tex] is [tex]\(\boxed{-0.7041}\)[/tex].